拓扑电路入门学习

本文打算Follow武汉大学余睿老师的综述 电路中的拓扑态 , 以及他的讲座 在电路中设计新拓扑物态 一些关键的概念需要掌握: 线性电子线路 电路理论(电势运动方程,拉普拉斯矩阵方程) 基尔霍夫方程与TB模型的对应(Onsite energy, hopping, SOC),基尔霍夫方程的求解, 具体Hspice的模拟我们另开新帖。

概念的阐释

线性电路

线性电路是指完全由线性元件、独立源或线性受控源构成的电路。(来自百度百科 )

Linear circuit : an electronic circuit which obeys the superposition principle(for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.) F(ax1+bx2)=aF(x1)+bF(x2)F\left(a x_1+b x_2\right)=a F\left(x_1\right)+b F\left(x_2\right)

线性元件

  • ideal resistors
  • ideal capacitors
  • ideal inductors
  • op-amps(operational amplifier)

非线性元件

线性电路的组成单元为线性元件(An element is linear if and only if the terminal voltage v and terminal current i, together with the initial condition, if any, satisfy the homogeneity and additivity properties in the equation that defines the element,就是元件满足线性欧姆定律):

常见线性电路

  • 放大电路
  • 积分电路
  • 差分电路
  • 滤波器

电势运动方程

并没有明确告诉我们电势运动方程是什么,猜测是以基尔霍夫电势方程为基石的运动方程 The fundamental assumption of circuit theory is that the voltages satisfy Kirchhoff’s voltage law (KVL) and the currents satisfy Kirchhoff’s current law (KCL).

  • KVL
  • KCL

拉普拉斯矩阵方程

基尔霍夫方程与TB模型的对应1

a little example

mesh current method

Complex currents J~1\tilde{J}_1 and J~2\tilde{J}_2 circulate along the two directed meshes α1\alpha_1 and α2\alpha_2. V~1\tilde{V}_1 (Q and G)and V~2\tilde{V}_2

Hereby: We can represent them by the vector J~=[J~1,J~2]T\tilde{\boldsymbol{J}}=\left[\tilde{J}_1,\tilde{J}_2\right]^{T}. The voltage drops along α1,α2\alpha_1, \alpha_2 are denoted by V~=[V~1,V~2]T\tilde{\boldsymbol{V}}=\left[\tilde{V}_1,\tilde{V}_2\right]^{T}. The voltage drop vector V~\tilde{\boldsymbol{V}} and the mesh current vector J~\tilde{J} are related as V~=Z(ω)J~\tilde{\boldsymbol{V}}=Z(\omega) \tilde{\boldsymbol{J}}

Z(ω)=[jωL+1/(jωCt)1/(jωC)1/(jωC)jωL+1/(jωCt)] Z(\omega)=\left[\begin{array}{cc} \mathrm{j} \omega L+1 /\left(\mathrm{j} \omega C_{\mathrm{t}}\right) & -1 /\left(\mathrm{j} \omega C^{\prime}\right) \newline -1 /\left(\mathrm{j} \omega C^{\prime}\right) & \mathrm{j} \omega L+1 /\left(\mathrm{j} \omega C_{\mathrm{t}}\right) \end{array}\right]

Due to the KVL, V~\tilde{\boldsymbol{V}} must be zero, We have detZ(ω)=0detZ(\omega) = 0

electrical node potentials

Y(ω)=[jωC+1/(jωL)jωC0jωC2jωC+jωCjωC0jωCjωC+1/(jωL)] Y(\omega)=\left[\begin{array}{ccc} \mathrm{j} \omega C+1 /(\mathrm{j} \omega L) & -\mathrm{j} \omega C & 0 \newline -\mathrm{j} \omega C & 2 \mathrm{j} \omega C+\mathrm{j} \omega C^{\prime} & -\mathrm{j} \omega C \newline 0 & -\mathrm{j} \omega C & \mathrm{j} \omega C+1 /(\mathrm{j} \omega L) \end{array}\right]

MNA

General analysis

the resonance condition and show the analogy between electrical circuits and quantum tight-binding models in solid-state physics.

Resonant circuits with inductances

Resonant circuits with mutual inductances

For this circuit network, Z^(ω)=i(jωLi+1jωC)αiαi+jωiiiMijαiαj \begin{aligned} \hat{Z}(\omega)=& \sum_{i}\left(\mathrm{j} \omega L_{i}+\frac{1}{\mathrm{j} \omega C}\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right| \newline &+\mathrm{j} \omega \sum_{i} \sum_{i \neq i} M_{i j}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

then jωCZ^Ψ>=0j\omega C\hat{Z}|\Psi\rangle> = 0

jωCZ^(ω)=i(ξij2(ω/ω0)2+1)αiαi+j2ω2CiiiMijαiαj \begin{aligned} j\omega C \hat{Z}(\omega)=& \sum_{i}\left( \xi_{i} \mathrm{j}^2 (\omega/\omega_0)^2 +1\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right| \newline &+\mathrm{j}^2 \omega^2 C \sum_{i} \sum_{i \neq i} M_{i j}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

then (ω0ωj)2jωCZ^Ψ=0(\frac{\omega_0}{\omega j})^2j\omega C\hat{Z}|\Psi\rangle = 0

(ω0ωj)2jωCZ^(ω)=i(ξi+(ω0ω)2)αiαi+(ω2/ω02)/LiiiMijαiαj \begin{aligned} (\frac{\omega_0}{\omega j})^2 j\omega C \hat{Z}(\omega)=& \sum_{i}\left( \xi_{i} +(\frac{\omega_0}{\omega})^2\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right| \newline &+ \sout{(\omega^2/\omega_0^2)}/L \sum_{i} \sum_{i \neq i} M_{i j}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

We have

K^Ψ=(ω0ω)2ΨK^=iξiαiαiijiκijαiαj \begin{aligned} &\hat{K}|\Psi\rangle=\left(\frac{\omega_{0}}{\omega}\right)^{2}|\Psi\rangle \newline &\hat{K}=\sum_{i} \xi_{i}|\alpha_{i}\rangle\langle\alpha_{i}|-\sum_{i} \sum_{j \neq i} \kappa_{i j}| \alpha_{i}\rangle\langle\alpha_{j}| \end{aligned}

where ω0=1/LC\omega_0 = 1/\sqrt{LC}, ξi=Li/L\xi_{i}=L_i/L and κij=Mij/L\kappa_{ij} = - M_{ij}/L

Then K^\hat{K} can be interpreted as a Hamiltonian: K^Ψ==H^Ψ=EΨ=(ω0ω)2Ψ\hat{K}|\Psi\rangle= = \hat{H}|\Psi\rangle = E|\Psi\rangle=\left(\frac{\omega_{0}}{\omega}\right)^{2}|\Psi\rangle

Resonant circuits with coupling inductances

We now consider resonant circuits, composed of an inductance Li and a capacitance C connected in parallel to π\pii, as shown in Fig. 2(b). Y^(ω)=i(1jωLi+jωC)πiπi+ijωLij1jωLij(πiπiπiπj) \begin{aligned} \hat{Y}(\omega)=& \sum_{i}\left(\frac{1}{j \omega L_{i}}+\mathrm{j} \omega C\right)\left|\pi_{i}\right\rangle\left\langle\pi_{i}\right| \newline &+\sum_{i} \sum_{j \omega L_{i j}}\frac{1}{j \omega L_{ij}}\left(\left|\pi_{i}\right\rangle\left\langle\pi_{i}|-| \pi_{i}\right\rangle\left\langle\pi_{j}\right|\right) \end{aligned}

Introducing an inductance L for nondimensionalization, we rewrite the equation (ω0ωj)2jωLY^Ψ=0(\frac{\omega_0}{\omega j})^2j\omega L\hat{Y}|\Psi\rangle = 0 as follows: K^Ψ=(ωω0)2Ψ \hat{K}|\Psi\rangle=\left(\frac{\omega}{\omega_0}\right)^{2}|\Psi\rangle

1-D example

1-D Resonant circuits with mutual inductances

Z^(ω)=(jωL+1jωC)1^+jωMiZ(αi1αi+αi+1αi) \hat{Z}(\omega)=\left(j \omega L+\frac{1}{j \omega C}\right) \hat{1}+j \omega M \sum_{i \in \mathbb{Z}}\left(\left|\alpha_{i-1}\right\rangle\left\langle\alpha_{i}|+| \alpha_{i+1}\right\rangle\left\langle\alpha_{i}\right|\right)

Here, we construct K^(p)=iξ0αiα0\hat{K}^{(\mathrm{p})}=\sum_{i} \xi_{0}\left|\alpha_{i}\right\rangle\left\langle\alpha_{0}\right|, ξ0=1\xi_0= 1, K^(h)=ijiκαiαj\hat{K}^{(\mathrm{h})}=-\sum_{i} \sum_{j \neq i} \kappa \left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right|

H(k)=ξ0+eikΔxκ+eikΔxκH(k) =\xi_{0} + e^{ik\Delta x}\kappa + e^{-ik\Delta x}\kappa

H(k)=ξ0+2cos(kΔx)κ=(ω0ω)2H(k) = \xi_{0}+ 2cos(k\Delta x)\kappa = (\frac{\omega_{0}}{\omega})^2

Resonant circuits with capacitance

对电路网络中的节点进行编号 α=1,2,,N\alpha = 1, 2, · · · , N , 任意两节点 a 和 b 间导纳记为yαβy_\alpha\beta. 节点a上的电势记为vαv_\alpha(取地面为零势能参考点), 流入该节点的净电流记为 IαI_\alpha. 对节点α\alpha , 从与之相连的节点α\alpha^{\prime}流入的电流记为IααI_\alpha\alpha^{\prime}, 欧姆定律要求 Iαα=(vαvα)yααI_\alpha\alpha^{\prime} = (v_\alpha^{\prime} - v_\alpha) y_\alpha\alpha^{\prime} , 求和

α(vαvα)yαα=Iα\sum_{\alpha^{\prime}}(v_\alpha^{\prime} - v_\alpha) y_\alpha\alpha^{\prime} = I_\alpha

LV=I\mathcal{L} V = I

这个时候我们把hopping项换成电容, 利用 孤立的 LC 谐振回路具有特定的谐振频率 ω0\omega_0 = 1/LC1/\sqrt{LC}

The admittance matrix Use Y(ω)Y(\omega), 导纳 WaveVector-> Voltage

Y^(ω)=i(1jωLi+jωC)πiπi+ijωCijjωCij(πiπiπiπj) \begin{aligned} \hat{Y}(\omega)=& \sum_{i}\left(\frac{1}{j \omega L_{i}}+\mathrm{j} \omega C\right)\left|\pi_{i}\right\rangle\left\langle\pi_{i}\right| + \sum_{i} \sum_{j \omega C_{i j}}j\omega C_{ij}\left(\left|\pi_{i}\right\rangle\left\langle\pi_{i}|-| \pi_{i}\right\rangle\left\langle\pi_{j}\right|\right) \end{aligned}

1jωCY^(ω)=i(ω02ω2+ci)πiπi+icijcij(πiπiπiπj) \begin{equation} \begin{aligned} \frac{1}{j\omega C}\hat{Y}(\omega)=& \sum_{i}\left(-\frac{\omega_0^2}{ \omega^2 }+ c_i\right)\left|\pi_{i}\right\rangle\left\langle\pi_{i}\right| + \sum_{i} \sum_{c_{ij}}c_{ij}\left(\left|\pi_{i}\right\rangle\left\langle\pi_{i}|-| \pi_{i}\right\rangle\left\langle\pi_{j}\right|\right) \end{aligned} \end{equation}

1jωY^(ω)=i(1ω2L+Ci)πiπi+iCijCij(πiπiπiπj) \begin{aligned} \frac{1}{j\omega}\hat{Y}(\omega)=& \sum_{i}\left(-\frac{1}{ \omega^2 L }+ C_i\right)\left|\pi_{i}\right\rangle\left\langle\pi_{i}\right| + \sum_{i} \sum_{C_{ij}}C_{ij}\left(\left|\pi_{i}\right\rangle\left\langle\pi_{i}|-| \pi_{i}\right\rangle\left\langle\pi_{j}\right|\right) \end{aligned}

YV=1ω2LV,Y=[y1C12C1NC21y2C2NCN1CN2yN]yα=α=1(α)NCαα+Cα \begin{equation} \begin{aligned} &Y V=\frac{1}{\omega^{2} L} \boldsymbol{V}, \newline &Y=\left[\begin{array}{cccc} y_{1} & -C_{12} & \cdots & -C_{1 N} \newline -C_{21} & y_{2} & \cdots & -C_{2 N} \newline \vdots & \vdots & \ddots & \vdots \newline -C_{N 1} & -C_{N 2} & \cdots & y_{N} \end{array}\right] \newline &y_{\alpha}=\sum_{\alpha^{\prime}=1(\neq \alpha)}^{N} C_{\alpha \alpha^{\prime}}+C_{\alpha} \end{aligned} \end{equation}

impedance matrix Use Z(ω)Z(\omega), 阻抗 WaveVector-> Current

For this circuit network, Z^(ω)=i(jωL+1jωCi)αiαi+1jωiii1Cijαiαj \begin{aligned} \hat{Z}(\omega)=& \sum_{i}\left(\mathrm{j} \omega L+\frac{1}{\mathrm{j} \omega C_{i}}\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right|+\frac{1}{\mathrm{j} \omega} \sum_{i} \sum_{i \neq i} \frac{1}{C_{i j}}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

1jωLZ^(ω)=i(1ω02ω2ϵi)αiαiiiiω02ω2ϵijαiαj \begin{aligned} \frac{1}{\mathrm{j} \omega L}\hat{Z}(\omega)= & \sum_{i}\left(1-\frac{\omega_0^2}{\omega^2 \epsilon_{i}}\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right|-\sum_{i} \sum_{i \neq i} \frac{\omega_0^2}{\omega^2\epsilon_{i j}}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

(ωω0)21jωLZ^(ω)=i((ωω0)2+1ϵi)αiαi+iii1ϵijαiαj \begin{aligned} -(\frac{\omega}{\omega_0})^2\frac{1}{\mathrm{j} \omega L}\hat{Z}(\omega)= & \sum_{i}\left(-(\frac{\omega}{\omega_0})^2+ \frac{1}{\epsilon_{i}}\right)\left|\alpha_{i}\right\rangle\left\langle\alpha_{i}\right|+\sum_{i} \sum_{i \neq i} \frac{1}{\epsilon_{i j}}\left|\alpha_{i}\right\rangle\left\langle\alpha_{j}\right| \end{aligned}

所以我们有 H^circuV=(ωω0)2V\hat{H}_{circu}\bf{V} =(\frac{\omega}{\omega_0})^2\bf{V}

H^circu=i1ϵiαiαi+iji1ϵijαiαj \hat{H}_{circu} = \sum_{i} \frac{1}{\epsilon_{i}}|\alpha_{i}\rangle\langle\alpha_{i}| +\sum_{i} \sum_{j \neq i} \frac{1}{\epsilon_{ij}}|\alpha_{i}\rangle\langle\alpha_{j}|

拓扑电路领域的关键发展


  1. Y. Nakata, T. Okada, T. Nakanishi, and M. Kitano, Circuit Model for Hybridization Modes in Metamaterials and Its Analogy to the Quantum Tight-Binding Model: Circuit Model for Hybridization Modes in Metamaterials, Phys. Status Solidi B 249, 2293 (2012). ↩︎


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